Some intimate links between integrable systems and combinatorics have been revealed in recent years. An interesting observation is that many discrete integrable systems exhibit the so-called Laurent phenomenon, and many mappings with the Laurent property are proved to be integrable. The Laurent property is usually proved by cluster algebra in the literature. Recently, we investigated three discrete integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called -system and derived their general solutions in terms of Hankel determinant. From their determinant solutions, we obtained many properties including the Laurent property. By using Hirota's bilinear approach, we also indicated that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. Additionally, a conjecture based on Somos-4 (P. Barry, J. Integer Seq. 13 (2010); P. Barry, arXiv:1107.5490 (2011)) was confirmed.